The two angles of the triangle are 60 and 20 degrees. a) determine in what ratio the vertices of the triangle

The two angles of the triangle are 60 and 20 degrees. a) determine in what ratio the vertices of the triangle divide the circumscribed circle. b) find the angles of the triangle, the vertices of which are the points of tangency of the inscribed circle with the sides of this triangle.

Known 2 angles of the triangle.

<1 = 60 °;

<2 = 20 °;

Find the third corner of the triangle.

The sum of all the angles of a triangle is 180 °.

<3 = 180 ° – <1 – <2 = 180 ° – 60 ° – 20 ° = 120 ° – 20 ° = 100 °.

a) The ratio of the vertices of the triangle that divide the circle.

20 °: 60 °: 100 ° = 1: 3: 5;

b) <1 = 1/2 * (<2 + <3) = 1/2 * (20 ° + 100 °) = 1/2 * 120 ° = 120 ° / 2 = 60 °;

<2 = 1/2 * (<1 + <3) = 1/2 * (60 ° + 100 °) = 1/2 * 160 ° = 160 ° / 2 = 80 °;

<3 = 1/2 * (<1 + <2) = 1/2 * (20 ° + 60 °) = 1/2 * 80 ° = 80 ° / 2 = 40 °.